## Natural and smoothing quadratic spline. (An elementary approach).(English)Zbl 0731.65006

The authors study the problem of constructing a parabolic spline s on [a,b] with knots $$\{x_ i\}^ n_ 1$$, $$a=x_ 0<x_ 1<...<x_{n+1}=b$$, satisfying the interpolation conditions (1) $$\int^{x_{i+1}}_{x_ i}s(t)dt=y_ i,$$ $$i=0,1,...,n$$. They show that the unique solution s (under certain boundary conditions) minimizes the integral $$\int^{b}_{a}[s'(t)]^ 2dt$$ over the set of all interpolants from $$W^ 1_ 2[a,b].$$
Reviewer’s remark: Set $$p'(t)=s(t)$$, $$p(a)=0$$. Then the problem (1) is equivalent to the cubic spline interpolation problem $$p(x_ i)=f_ i,$$ $$i=0,1,...,n+1$$, with $$f_ 0=0$$, $$f_ i:=y_ 0+y_ 1+...+y_{i-1}$$, $$i=1,...,n+1$$, and many of the results in this paper follow from the known theorems about natural spline interpolation.

### MSC:

 65D07 Numerical computation using splines 41A15 Spline approximation 65D05 Numerical interpolation
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### References:

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